Snow Ploughs

A lovely problem, taken from Advanced Problems in Mathematics, by Dr Siklos*

Two identical snowploughs plough the same stretch of road. The first starts at a time

seconds after it starts snowing, and the second starts from the same point seconds later, going in the same direction. Snow falls so that the depth of snow increases at a constant rate . The speed of each snowplough is , where is the depth (in metres) of the snow it is ploughing, and is a constant. Each snowplough clears all the snow. Show that the time at which the second snowplough has travelled a distance metres satisfies the equation: . Hence show that the snowploughs will collide when they have travelled metres.

*He adds in his discussion that this can be generalised to n identical snowploughs

Re: Snow Ploughs

When the first snow plough has been moving for seconds, the depth of snow at the spot it has reached is metres so its velocity would be metres per second. The distance it has moved out would thus be metres.